Question: Plates and Bowls

Comment on Plates and Bowls

I tried using a system of two equestions for this question.
p+b=100. Then i used 11(p)+5(b)=100. I tried combining the two to get rid of a variable but i still didnt get a clean integer. Is this not a valid approach? I just didnt think to set up a table. Should consider it next time.
gmat-admin's picture

Good try, but there's a problem with the equation: p + b = 100

In your first equation, 11p + 5b = 100, the variable p represents the NUMBER of plates purchased, and the variable b represents the NUMBER of bowls purchased.

So, p + b must represent the total NUMBER of plates AND bowls purchased. In your equation, you have p + b = 100, which suggests that Bart a total of 100 plates and bowls, which is not true. In the given information, 100 is the total amount of money that Bart spent, and you correctly used that number in the equation 11p + 5b = 100

So, using statement 1, you can only create one equation (11p + 5b = 100). However, as we can see in the video solution, this equation is sufficient to answer the target question with certainty.

in statement 1 through algebraic approach how could we solve that equation with two variables p & B.??
gmat-admin's picture

Let p = the NUMBER of plates purchased
Let b = the NUMBER of bowls purchased

If plates cost $11 each and bowls cost $5 each, then we can write: 11p + 5b = 100 (since the total cost is $100)

IMPORTANT: In most cases, the equation 11p + 5b = 100 has infinitely many solutions. However, with real-world questions like this, p and b must be POSITIVE INTEGERS. It turns out that this huge restriction limits the number of solutions to the equation 11p + 5b = 100. In fact, as you can see in the video, there's only ONE solution that meets the given conditions.

Hello, I tried to solve it using equation and reached conclusion that both statement together are sufficient. First statement gives us 11p+5b=100, I thought it has two variable so we can't solve

Second statement gives us b=p+4

by solving these two statement we can get p=5 and b=9

so if I don't think of using table method I can simply end up with wrong option, how to avoid this mistake
gmat-admin's picture

This is a common trap on the GMAT, so it's important to be aware of situations in which an equation with two variables (like 11p + 5b = 100) does NOT have infinitely many solutions.

If it were the case, that there were no restrictions on p and b, then there would be infinitely many solutions. Some solutions include p=9 & b=0.2, p=10 & b=-1, and p=1/11 & b=19.8 and so on.

However, since p and b must be POSITIVE INTEGERS, we need to actually confirm whether there is more than 1 solution before concluding that the statement is not sufficient.

I just had one doubt that while checking for statement 1, how can be rule out the possibility that he bought 0 plates and 20 bowls?
gmat-admin's picture

Great question!
The key word is "some."
On the GMAT, "some" means one or more.

This is the 3rd time I see similar question. my approach is to find a number between 1 and 100 inclusive that result in an integer when divided by 11&5. the only number is 55, thus statement is sufficient.

I found data sufficiency very hard and it starts to be a frustrating topic for me. Each time I think I got it, there is an info I didn't evaluate. In this example, I was sure C was going to be the answer as I thought that with 2 unknowns it would be impossible to solve the equation in statement 1. I didn't think at all about the positive integer deduction.

Do you have tips or a sort of mental check list to go through in order to avoid those traps?

gmat-admin's picture

You're not alone. A lot of students believe that an equation with two variables can never be solved.

This myth, along with others, are mentioned in the following videos (Common GMAT Data Sufficiency Myths - Parts I & II):
Awareness of the above myths is the best way to avoid falling for this trap.

If a linear equation has two variables, and you need to find the value of 1 (or both) variable(s), you should first ask "Are the variables restricted to positive integers?"
- If the variables are restricted to positive integers, then there MAY be just one solution.
- If the variables are not restricted (i.e., they can be any real numbers), then there will be infinitely many solutions.

Here's an official question to practice with:

I see this immediately --number sense!

two must be integers so only if there are 5 P can B be 45, which is divisible by 5 to becomes 9
gmat-admin's picture

Very nice!

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